The basic idea is that a small group of people, that were infected by the virus but managed to survive, has their hair colour changed to white and their eye colour changed to green. Said traits would then be passed on to future generations.

Some context info (changeable if needed):

1 The affected group is ethnically pure = physical traits vary less (think modern-day Japan).

2 Eye colour has changed from blue to green; if not green then gray.

3 These traits aren't dominant (this comes from my very crude understanding of genetics); a kid would need both of his/her parents to be affected by the virus to be fully white-haired/green-eyed.

4 The virus has a kill-rate of 90%.

5 The world is comparable to Europe's late Medieval Era.

6 There's no magic.

(Edited):

7 The virus is transmitted through the air.

8 It is naturally occurring.

(1) Ebola

• 
  

  Changed one doctor's eye color from blue to green. Although in this specific case his eye color did change back, the article notes:

  
  
  

  
  

  Though it is quite rare for eye color to change so dramatically, this does happen from time to time as a result of viral infections and is usually permanent. Changes in color are usually due to the viral infection damaging pigmented cells in the iris. Following treatment, however, Crozier’s eye returned to normal, though it remains unclear why.

  
  

  
  



• There are links between viral infections and graying hair color



• Had a kill rate as high as 89% kill rate in the Democratic Republic of the Congo
  


• Although Ebola doesn't pass on genetically, in a medieval-type world without proper sanitation (or even awareness of biotics) it could easily spread to children, especially if one or both parents had the disease.


• If your story included a mutated form of Ebola that attached to people genetically, the other criteria make sense.

(2) Shingles

• 
  Shingles is a crazy virus (a form of herpes) that often lays dormant for many years
  


• can cause severe hearing loss or blindness. In rare cases, can lead to death


• again, by default it doesn't pass on genetically, but (a) is part of the same virus that causes chickenpox, which was considered "norm" for children due to its prevalence until vaccines were made for it. And (b) a mutated form of Shingles would not be difficult to image, one that attached (passively) to children during insemination, but laid dormant.


• although mortality rates of shingles is low, a mutated form could be more deadly. Given its tendency to lay dormant without detection or activation for many years, it's unlikely a medieval-type society would be able to prevent it.

There are lots of things that can depigment hair and skin. Some drugs can do it, but that is reversible. Age of course can do it. Vitiligo is an autoimmune condition; affected skin loses pigment and hair coming from that skin can be white. It is possible to develop vitiligo because you have melanoma - an immune attack on the pigmented tumor cells also attacks pigmented non tumor cells. I do not know (until just now) of any virus which leaves white scars.

Iris depigmentation is rare. Drugs that depigment hair and skin don't affect the iris, nor does age (not counting arcus senilis) Vitiligo does not usually affect the iris. But viral infection apparently can cause depigmentation of the iris.

Bilateral acute depigmentation of the iris: report of 26 new cases and four-year follow-up of two patients.

https://aibolita.com/eye-diseases/38504-gregory-ostrow-md.html

As I understand it this is not an immune attack on the pigment cells but direct infection and destruction of those cells by virus.

So yes, your virus could cause lasting pigment change.. Have the melanocytes be one of the targets of this virus such that they are destroyed throughout the body. Or have an immune attack on the virus that hits melanocytes. Or both.

Genetically transferred virus

User cegfault did a good job showing that Ebola would be a good candidate virus for killing 90% of the infected and causing grey hair and green eyes. But Ebola is not passed on genetically. BUT there are ways viruses could be passed on genetically. Here's how:

Retroviruses insert themselves into the DNA of our cells, but we don't pass on most cells to our children (e.g. chickenpox lives in nerve cells not in sperm/eggs). However, very rarely, viruses do get into our sex cells and so get passed on (when I mean rarely I mean back when our ancestors were lemurs). We know this has happened before and some scientists think it may be behind multiple sclerosis and schizophrenia (this is a super cool theory by the way, basically this virus can get activated during an illness and depending on how our body responds we can get MS or schizophrenia). Now its believed that these viruses accidentally ended up in that ancestor's sperm/egg cells but we could imagine a virus that preferentially inserted its DNA into sperm/eggs just like chickenpox hangs out in nerve cells.

Ok great, now we've got a virus in the parents sperm/egg DNA but still need to inherit that virus from both parents to have the change in hair and eye color. Well, if we inherit one "healthy" side of DNA that DNA continues to make the correct eye color, perhaps even compensating for the "bad" DNA (just like how carriers of albinism aren't lighter skinned but if they have children with another carrier 25% of the children will be albino). The only problem so far is that Ebola is not a retrovirus so cant insert itself into your cells (it only uses RNA not DNA). But perhaps through bacteria sex a retrovirus could obtain those attributes of Ebola that we are interested in. Voila, now you can get the virus from your parents or from the air (and once you get it from the air it inserts itself into your sex cells and you pass it on to your kids). Now why doesn't the virus from your mum's cells infect the cells with your dad's DNA? Perhaps the cells only insert their DNA into the sex cells.

A genetically engineered retrovirus is fully capable of changing genetic traits, including eye and hair colour, but as your setting is medieval, I doubt anyone is engineering viruses.